Question

Statement- 1: A person purchase \(1 \mathrm{~kg}\) of tomatoes from each of the 4 places at the rate of \(1 \mathrm{~kg}, 2 \mathrm{~kg}, 3 \mathrm{~kg}, 4 \mathrm{~kg}\) per rupee respectively. On the average he has purchased \(x \mathrm{~kg}\) of tomatoes per rupee, then the value of \(x\) is \(2.5\). Statement- \(2:\) The average of \(x_{1}, x_{2}, x_{3}, \ldots \ldots, x_{n}\) is \(\frac{x_{1}+x_{2}+x_{3}+\ldots+x_{n}}{n}\) (1) Statement \(-1\) is True, Statement \(-2\) is True; Statement \(-2\) is a correct explanation for Statement \(-1\) (2) Statement- 1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1 (3) Statement \(-1\) is True, Statement - 2 is False (4) Statement - 1 is False, Statement -2 is True

Short answer

Statement 1 is False, Statement 2 is True. Option (4) is correct.

Step-by-step solution

Understanding the Problem

A person buys 1 kg of tomatoes from each of 4 places with the rates 1 kg, 2 kg, 3 kg, and 4 kg per rupee respectively. We need to determine the average kilograms of tomatoes purchased per rupee and verify the given statements.

Calculate Total Tomatoes Purchased

The person buys 1 kg from each place, so the total tomatoes purchased from the four places is: 1 kg + 1 kg + 1 kg + 1 kg = 4 kg.

Calculate Total Rupees Spent

The person buys 1 kg from place 1 (1 kg per rupee), which costs 1 rupee. For place 2 (2 kg per rupee), 1 kg costs 0.5 rupees. For place 3 (3 kg per rupee), 1 kg costs approximately 0.33 rupees. For place 4 (4 kg per rupee), 1 kg costs 0.25 rupees. Therefore, the total rupees spent = 1 + 0.5 + 0.33 + 0.25 = 2.08 rupees (approximately).

Calculate Average Kilograms per Rupee

The average kilograms of tomatoes purchased per rupee is the total kilograms divided by the total rupees spent: \ \ \( x = \frac{4 \text{ kg}}{2.08 \text{ rupees}} \approx 1.92 \text{ kg per rupee} \).

Analyze Statement 1

Statement 1 says that the average is 2.5 kg per rupee, which is higher than our calculation of 1.92 kg per rupee. Therefore, Statement 1 is False.

Analyze Statement 2

Statement 2 gives the formula for calculating the average of a set of numbers, which is correct. Therefore, Statement 2 is True.

Conclusion

Based on the analysis, Statement 1 is False and Statement 2 is True. The correct answer is option (4).

Key concepts

average calculations

Average calculations are a fundamental aspect of mathematics. They help determine a central value from a set of numbers. In simple terms, the average is the sum of the numbers divided by the count of numbers. For instance, if you want to find the average of 4, 5, and 6, you add these numbers to get 15. Then, you divide 15 by 3, resulting in an average of 5. This concept is crucial in various scenarios, from academic problems to real-life applications.

Consider the exercise with tomatoes. To find the average kilograms of tomatoes per rupee, we summed up the tomatoes bought at different rates and divided by the total rupees spent. We found that the average was not 2.5 kg per rupee as stated, but approximately 1.92 kg per rupee. This miscalculation highlights the importance of accuracy in average calculations.

Always remember:

• Sum the values.
• Count the number of values.
• Divide the sum by the count.

mathematical problem solving

Mathematical problem solving is like being a detective. It involves understanding the problem, planning a way to tackle it, solving it step-by-step, and then checking the solution. Let's break down the process using the exercise about the tomatoes:

• Understanding the Problem: The person buys tomatoes at different rates. We need to find the average kilograms per rupee spent.
• Planning: Identify the total tomatoes bought and the total rupees spent.
• Solving: Calculate total tomatoes and total rupees. Then, find the average.
• Checking: Verify if the calculated average matches the given statements.

This approach ensures that we systematically arrive at the correct answer. Using this structure helps in minimizing mistakes and achieving a clear understanding of the problem. So, whenever you face a mathematical problem, think like a detective and follow these steps.

IIT JEE preparation

Preparing for IIT JEE is a journey that requires strong foundational knowledge and strategic study methods. Here's how you can excel:

First, focus on understanding core concepts, like average calculations. They often form the basics for more complex problems.

Consistent practice is key. Solve a variety of problems regularly to build and maintain your skills.

Study solutions step-by-step. Break down the solutions, as we did with the tomato exercise, to understand each part clearly.

Take mock tests and time yourself. This will help you manage time effectively during the actual exam.

• Review your mistakes.
• Understand where you went wrong.
• Learn the correct method.

Finally, don't hesitate to seek help. Use platforms, join study groups, and ask questions whenever you're stuck. Remember, persistence and patience are crucial. Stay motivated and keep learning!